Consider the random variables number of occurences of a factor (consecutiv\ e subword) in a random n-letter word in the alphabet {1,2},of 11 and 22 The average and variance for 11 are 5 n [- 1/4 + n/4, - 7/16 + ---] 16 The average and variance for 22 are 5 n [- 1/4 + n/4, - 7/16 + ---] 16 4 The correlation is, - 3/5 + ---- 25 n The asymptotics for the 8^2 mixed moments are 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 6 16 5 3 16 5 12 16 5 42 16 5 ------------ + ------------ ------------- - ------------- 25 125 n 62 5 25 n [[1, 0, 1, ---------------------------, 3 + ----, -----------------------------, 1/2 25 n 1/2 n n 1/2 1/2 1/2 1/2 126 16 5 2121 16 5 -------------- - --------------- 2082 5 125 n 45276 4 15 + ----, --------------------------------, 105 + -----], [0, - 3/5 + ----, 25 n 1/2 25 n 25 n n 1/2 1/2 1/2 1/2 2 16 5 16 5 - ------------ - ---------- 25 125 n 42 ---------------------------, - 9/5 + ----, 1/2 25 n n 1/2 1/2 1/2 1/2 132 16 5 214 16 5 - -------------- + -------------- 125 125 n 42 ---------------------------------, -9 - ---, 1/2 5 n n 1/2 1/2 1/2 1/2 306 16 5 4467 16 5 - -------------- + --------------- 25 125 n 66192 ----------------------------------, -63 - -----, 1/2 125 n n 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 3864 16 5 271572 16 5 2 16 5 16 5 - --------------- + ----------------- - ------------ - ---------- 25 625 n 25 125 n -------------------------------------], [1, ---------------------------, 1/2 1/2 n n 1/2 1/2 1/2 1/2 36 16 5 86 16 5 ------------- - ------------- 43 378 125 125 n 183 3062 -- - -----, -----------------------------, --- - -----, 25 125 n 1/2 25 125 n n 1/2 1/2 1/2 1/2 726 16 5 17913 16 5 -------------- - ---------------- 125 625 n 5988 ---------------------------------, 237/5 - -----, 1/2 125 n n 1/2 1/2 1/2 1/2 10416 16 5 351288 16 5 ---------------- - ----------------- 125 625 n 2163588 ------------------------------------, 2037/5 + -------], [ 1/2 625 n n 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 6 16 5 3 16 5 36 16 5 86 16 5 ------------ + ------------ ------------- - ------------- 25 125 n 42 125 125 n ---------------------------, - 9/5 + ----, -----------------------------, 1/2 25 n 1/2 n n 1/2 1/2 1/2 1/2 1098 16 5 32919 16 5 - --------------- + ---------------- 837 20718 625 3125 n 999 36432 - --- + -----, ------------------------------------, - --- + -----, 125 625 n 1/2 25 125 n n 1/2 1/2 1/2 1/2 26208 16 5 1268928 16 5 - ---------------- + ------------------ 625 3125 n 8127 891576 ---------------------------------------, - ---- + ------, 1/2 25 625 n n 1/2 1/2 1/2 1/2 443394 16 5 26762631 16 5 - ----------------- + ------------------- 625 3125 n 62 -----------------------------------------], [3 + ----, 1/2 25 n n 1/2 1/2 1/2 1/2 132 16 5 214 16 5 - -------------- + -------------- 125 125 n 183 3062 ---------------------------------, --- - -----, 1/2 25 125 n n 1/2 1/2 1/2 1/2 1098 16 5 32919 16 5 - --------------- + ---------------- 625 3125 n 23769 1125588 ------------------------------------, ----- - -------, 1/2 625 3125 n n 1/2 1/2 1/2 1/2 7992 16 5 450132 16 5 --------------- - ----------------- 625 3125 n 35757 453036 -----------------------------------, ----- - ------, 1/2 125 125 n n 1/2 1/2 1/2 1/2 1150758 16 5 91637637 16 5 ------------------ - ------------------- 3125 15625 n 347823 80676246 ----------------------------------------, ------ - --------], [ 1/2 125 3125 n n 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 12 16 5 42 16 5 726 16 5 17913 16 5 ------------- - ------------- -------------- - ---------------- 5 25 n 42 125 625 n -----------------------------, -9 - ---, ---------------------------------, 1/2 5 n 1/2 n n 1/2 1/2 1/2 1/2 7992 16 5 450132 16 5 --------------- - ----------------- 999 36432 625 3125 n 171207 2657736 - --- + -----, -----------------------------------, - ------ + -------, 25 125 n 1/2 625 625 n n 1/2 1/2 1/2 1/2 71514 16 5 6406011 16 5 - ---------------- + ------------------ 625 3125 n 1563597 30541518 ---------------------------------------, - ------- + --------, 1/2 625 625 n n 1/2 1/2 1/2 1/2 59799852 16 5 7065472554 16 5 - ------------------- + --------------------- 15625 78125 n 2082 ---------------------------------------------], [15 + ----, 1/2 25 n n 1/2 1/2 1/2 1/2 306 16 5 4467 16 5 - -------------- + --------------- 25 125 n 5988 ----------------------------------, 237/5 - -----, 1/2 125 n n 1/2 1/2 1/2 1/2 26208 16 5 1268928 16 5 - ---------------- + ------------------ 625 3125 n 35757 453036 ---------------------------------------, ----- - ------, 1/2 125 125 n n 1/2 1/2 1/2 1/2 71514 16 5 6406011 16 5 - ---------------- + ------------------ 625 3125 n 7551351 869022558 ---------------------------------------, ------- - ---------, 1/2 3125 15625 n n 1/2 1/2 1/2 1/2 18763164 16 5 2445294978 16 5 ------------------- - --------------------- 15625 78125 n 81004203 11282155794 -------------------------------------------, -------- - -----------], [ 1/2 3125 15625 n n 1/2 1/2 1/2 1/2 126 16 5 2121 16 5 -------------- - --------------- 5 125 n 66192 --------------------------------, -63 - -----, 1/2 125 n n 1/2 1/2 1/2 1/2 10416 16 5 351288 16 5 ---------------- - ----------------- 125 625 n 8127 891576 ------------------------------------, - ---- + ------, 1/2 25 625 n n 1/2 1/2 1/2 1/2 1150758 16 5 91637637 16 5 ------------------ - ------------------- 3125 15625 n 1563597 30541518 ----------------------------------------, - ------- + --------, 1/2 625 625 n n 1/2 1/2 1/2 1/2 18763164 16 5 2445294978 16 5 ------------------- - --------------------- 15625 78125 n 393117921 62870400306 -------------------------------------------, - --------- + -----------, 1/2 15625 78125 n n 1/2 1/2 1/2 1/2 1134058842 16 5 40450990923 16 5 - --------------------- + ---------------------- 78125 78125 n 45276 ------------------------------------------------], [105 + -----, 1/2 25 n n 1/2 1/2 1/2 1/2 3864 16 5 271572 16 5 - --------------- + ----------------- 25 625 n 2163588 -------------------------------------, 2037/5 + -------, 1/2 625 n n 1/2 1/2 1/2 1/2 443394 16 5 26762631 16 5 - ----------------- + ------------------- 625 3125 n 347823 80676246 -----------------------------------------, ------ - --------, 1/2 125 3125 n n 1/2 1/2 1/2 1/2 59799852 16 5 7065472554 16 5 - ------------------- + --------------------- 15625 78125 n 81004203 11282155794 ---------------------------------------------, -------- - -----------, 1/2 3125 15625 n n 1/2 1/2 1/2 1/2 1134058842 16 5 40450990923 16 5 - --------------------- + ---------------------- 78125 78125 n ------------------------------------------------, 1/2 n 23610565629 5007564767016 ----------- - -------------]] 78125 390625 n This took, 9.407, seconds.